Dynamics of delayed relay control systems with large delays

This paper investigates the dynamics of a piecewise linear delay differential equation modeling an inverted pendulum subject to delayed relay control. The inverted pendulum serves as an illustrative prototype example for an arbitrary saddle equilibrium. Delayed relay cannot give perfect stabilization of the equilibrium but generates small oscillations. On the other hand, one can construct simple switching manifolds that permit stable periodic orbits even with arbitrarily large delay in the control loop, provided the delay is known. Robustness of the stable periodic orbits with respect to parameter perturbations follows from their dynamical and structural stability in the dynamical systems sense.